A guide for a "Pirates of the Caribbean" role-playing game lists the following percentages for the rarity items the players may collect when a "loot box" is opened. Item rarity Percentage Super rare 5% Rare 10% Uncommon 25% Common 60% Sarah thinks that the guide is wrong. To test her suspicions, she opens 80| loot boxes. Her results are shown in the table below Item rarity Frequency Super rare 3 Rare 4 Uncommon 24 Common 49 Total 80 a) Calculate the expected frequency for each item. b) How many degrees of freedom are there in this test? c) Conduct a Chi-Squared test with a 1% level of significance to determine whether Sarah's suspicions are justified. USE A CRITICAL VALUE to make your decision
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
![**A Guide for "Pirates of the Caribbean" Role-Playing Game**
A guide for a "Pirates of the Caribbean" role-playing game lists the following percentages for the rarity items the players may collect when a "loot box" is opened.
| **Item Rarity** | **Super Rare** | **Rare** | **Uncommon** | **Common** |
|-----------------|----------------|----------|--------------|-------------|
| **Percentage** | 5% | 10% | 25% | 60% |
Sarah thinks that the guide is wrong. To test her suspicions, she opens 80 loot boxes. Her results are shown in the table below:
| **Item Rarity** | **Super Rare** | **Rare** | **Uncommon** | **Common** | **Total** |
|-----------------|----------------|----------|--------------|------------|-----------|
| **Frequency** | 3 | 4 | 24 | 49 | 80 |
### a) Calculate the expected frequency for each item.
To find the expected frequency for each rarity, calculate as follows:
- Super Rare: \(0.05 \times 80 = 4\)
- Rare: \(0.10 \times 80 = 8\)
- Uncommon: \(0.25 \times 80 = 20\)
- Common: \(0.60 \times 80 = 48\)
### b) How many degrees of freedom are there in this test?
The degrees of freedom for the Chi-Squared test is calculated as follows:
- Degrees of Freedom = (Number of Categories - 1)
- Here, Degrees of Freedom = \(4 - 1 = 3\)
### c) Conduct a Chi-Squared test with a 1% level of significance to determine whether Sarah’s suspicions are justified. USE A CRITICAL VALUE to make your decision.
1. **Observed Frequencies (O):** 3 (Super Rare), 4 (Rare), 24 (Uncommon), 49 (Common)
2. **Expected Frequencies (E):** 4 (Super Rare), 8 (Rare), 20 (Uncommon), 48 (Common)
3. **Chi-Squared Test Formula:**
\[
\chi^2 = \sum \frac{(O - E)^2}{E}
\]
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